Yet another way:

my $start_pattern = qr/.... $date/;
my $end_pattern = qr/.... $date/;
my $send_pattern = qr/.... whatever/;
while (<LOG>) {
  if (my $status = /$start_pattern/..1) {
    print "$. : $_"
      if $status == 1   or
         /$end_pattern/ or
         /$send_pattern/;
  }
}
[download]

Update: That should probably be (the first version only prints the start pattern once, this version will print it every time it occurs):

while (<LOG>) {
  if (/$start_pattern/..1) {
    print "$. : $_"
      if /$start_pattern/ or
         /$end_pattern/   or
         /$send_pattern/;
  }
}
[download]

Another update: After dropping the status, I'd go with an answer with a 'next unless' like dpuu's answer below.